Evans, who has a wild card for the main surgery in Wimbledon, returned to the top 200 after a few difficult years.
The English managed to nerve as the tension built against Paul. Evans achieved only 17 winners of America's 31, but also made 26 unintended mistakes compared to 43 Paul.
It will aim to be more clinical when the lucky American loser is facing Ginson Brooksby in the quarter -finals.
Earlier, Billy Harris Joy Estaborn continued the last with the English quarter -finals for the second year in a row.
Harris, who reached the main clouds as a lucky loser after losing in the qualifiers, defeated the Italian qualifiers Mattia Bellucci 6-3 6-4.
French Harris Ogu Hambert will face fourth, and Evans can meet in the semi -finals.
But the British number two Jacob Vernelli suffered a handicapped defeat by Marcus Jeron.
The American Fearnley 6-3 6-1 sent, as Scot hit five double errors and lost his service five times.
Wimbledon begins on June 30.
